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Chapter 5: Applications of Integration
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Section 5.3: Volume by Slicing
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Essentials


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•

The animation in Figure 5.3.1 shows a cutting plane intersecting a solid.

•

Suppose the slice at $x$ exposes a region $R\left(x\right)$ with area $A\left(x\right)$.

•

Further, suppose $R\left(x\right)$ is "thickened" by $\mathrm{dx}$ to a slab; the volume of this slab would be $A\left(x\right)\mathrm{dx}$.

•

The volume of the solid would be the sum of the volumes of the slabs, that is

$V\={\int}_{a}^{b}A\left(x\right)\mathit{DifferentialD;}x$

>

G:=proc(s)
local p1,p2,p3,p4;
p1:=plots:implicitplot3d((x/9)^2+(y/5)^2+(z/3)^2=1,x=9..9,y=5..5,z=3..3,scaling=constrained,style=surface,axes=frame,tickmarks=[0,0,0],labels=[x,"",""],transparency=.5,color=red):
p2:=plots:implicitplot3d(x=s,x=9..9,y=5..5,z=3..3,style=surface,color=green):
plots:display(p1,p2);
end proc:
plots:animate(G,[x],x=9..9,orientation=[50,70,0],frames=11,paraminfo=false);


Figure 5.3.1 Solid segmented by slicing






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Examples


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Example 5.3.1

By the method of slicing, obtain the volume of a wedge cut from a cylinder of radius $r$. In particular, let the axis of symmetry for the cylinder lie along the $z$axis, the bottom face of the wedge in the plane $z\=0$, and the slanted face of the wedge in the plane that passes through the origin and that makes an angle $\mathrm{\α}$ with the horizontal.

Example 5.3.2

By the method of slicing, obtain the volume of the solid whose base is an equilateral triangle of side $s$, and whose plane sections are squares. In particular, the equilateral triangle lies in the plane $z\=0$ and has a vertex at the origin, and an altitude along the $x$axis. The square cross sections are parallel to the $\mathrm{yz}$plane.

Example 5.3.3

By the method of slicing, obtain the volume of the solid whose base in the $\mathrm{xy}$plane is the region bounded by the $x$axis, and the curves $y\=\mathrm{sin}\left(x\right)$ and $x\=\mathrm{\π}\/2$, and whose cross sections parallel to the $\mathrm{yz}$plane are equilateral triangles.



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